The bird problem was a question on Victoria’s 1979 HSC Applied Mathematics exam. I raised the question in my 2020 Mathematics in Hell talk (at 35:45), in order to contrast it with the “hardest” 2019 Specialist Mathematics Exam 2 question (QB5). It is a nice problem (which I still haven’t solved) but, for the time, was just another nice problem. The bird problem hardly deserves its own post but the problem was raised again by frequent commenter Red Five on a recent post. So, what the hell. Maybe I’ll post some others from the era (and go back and polish off the TNDOT).
The bird problem was worth 10 marks, on an three hour exam with 160 marks in total. So, one can consider the bird problem being allotted about eleven minutes to solve. Acceleration due to gravity was taken to be 9.8 ms-2 and students were permitted (scientific) calculators. Red Five may be able to provide insight on the exam as a whole from the examiners’ report.
A stone is thrown so that it will hit a bird at the top of a pole. However, at the instant the stone is thrown, the bird flies away in a horizontal straight line at a speed of 10 ms-1. The stone reaches a height double that of the pole and, in its descent, touches the bird. Find the horizontal component of the velocity of the bird stone.
Hi. I think you meant the horizontal component of velocity of the stone here, or it is a very easy 10 marks 😊
Doh! That’s very funny. Thank you, Neil. Corrected.
The examiners’ report on this question simply stated (without giving an answer or solution): “This question was not done well.”.
I have my own solution but will allow others to ponder and post.
There are no tricks nor short-cuts I can find – just work carefully and methodically, giving your answer in exact form, perhaps?
Thanks, RF. I was interested in the report more generally.
As for the bird problem, I don’t think “exact form” is expected, although the question would be better as an exact form question (with g = 10 or g = g).
Yeah, I did it too – a few times – I thought there might be a super clean solution, but I couldn’t find one. Diagrams, thinking, simple motion formulas, algebra. 5(√2 + 1). No calc. But nice question.
Please don’t comment anonymously. But, very interesting. Is it obvious why? (I mean other than you divide both sides by g or whatever.)
You don’t need to use g at all. Just use the fact that orbit is a parabola. Turning things upside down, you can use y=ax^2. Then the constant a, and the height h, drop out when you do the simple calculation.
Very nice.
I agree about not needing
. My approach was to express 
in terms of 
(being the time the stone takes to reach the height of the bird and the time taken for the stone to complete its journey) and from there it all drops out nicely (enough).
I really like the problem but I wonder how 10 marks will be allocated. I started out thinking from kinematics then vector calculus angle of Specialist and made things rather complicated for myself. Probably I paid to mush attention to acceleration due to gravity. Then I back to the basic of a parabolic function with their features and no calculus required. A ha, an advance year 10 Maths should be inspired by this problems. A very nice way to introduce parameter at the lower level.
Birds
Thank you, Victoria. In the end a very neat solution, and presumably what Grant did. My guess is that the nice “it’s a parabola” approach was not intended, and was not common.
As for how the question was graded, that is a very good question. But I think on the whole, and the great value of having a 10-mark question, is that graders could more accurately reward good work, rather than dicking a kid three times for not including some pedantic twaddle. It would have, in a way, made the grading harder, and I am sure there was much greater variation in the grading. But there was much much less of the current madness.
There were 10 questions on this exam and, for comparison to today’s VCAA papers, each question was split into a maximum of 3 parts.
Each part was worth somewhere between 6 and 10 marks.
What a difference 50+ years makes.
I think 3or 4 such problems would be enough for spreading students on exam 2. No partitioning of problems, no scaffolding, the steps may be different for various methods.
N.B if different methods are to be promoted, imposing steps often ruins it.
Also, in partitioned problems too often solution to one part “resides” in other parts and I think students are punished for it, right or wrong.
Marking: entirely correct solution, no formal flaws, slips, key explanation: 10 marks, whatever method. There is room for marks deducted for minor offences, slips, inaccuracies.
Say 5 or 6 marks for the right method, explained/substantiated, but mucked up in the middle.
I think it would be important to allocate marking of each problem exclusively to a small group of markers , for consistency and correctness.
Not sure how it is done in Vic and other states.
Thanks, Polly.
Yes, one would not need the entire exam to be bird problems to enable reasonable discernment. Looking at the 1979 exam just now, there was a decent amount of pretty straight computational questions, although of a generally harder standard than now. And, yes, there was a lot more than “3 or 4” bird-ish questions.
As for the manner in which such questions should (or used to be) graded, it is an interesting and non-obvious question. Undoubtedly, the grading of a 10 mark question is more prone to subjective judgment and line calls, and it is reasonable to query whether the benefit of (re)introducing such questions is worth the cost of that subjectivity. The answer, for anyone who is not a complete idiot, is “Hell yes”, but I think the issue of subjectivity must reasonably be addressed.
The grading of the bird problem would reasonably fall along the lines you suggest, but more spelled out. Underlying the spelling out, you have a sense of what constitutes a “good try” and what that should be awarded: the 5 or 6 marks thing you point to. It would not be easy coordinating such VCE/HSC grading of a 10 mark question, but it is not *that* hard.
The other point, which teachers and students know all too well, and which is made incessantly on this blog, but which VCAA refuses to even consider, is that the current assessment system is “fair” in only a very narrow, technocratic sense. Students may be graded consistently on a question, but if the question and the grading is utterly insane, if it utterly fails to test what students should know, then the assessment is “fair” in only the narrowest and most artificial sense.
I could spend two minutes writing a marking scheme for the bird problem and it would be a million times fairer than what is happening now.